- #1
coalquay404
- 217
- 1
Hi. I've got a quick question on Penrose diagrams for the Schwarzschild space-time that I'd appreciate some comments on. In standard (t,r,\theta,\phi) coordinates the Schwarzschild metric is
ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2.
The immediate difficulty with this is that if one looks at radial null curves one finds that the light cones appear to close up as r\to 2M. The standard way of getting around this is to define a tortoise radial coordinate according to
r^* \equiv r + 2M\log|r-2M|.
Then the metric in (t,r^*,\theta,\phi) coordinates becomes simply
ds^2 = \left(1-\frac{2M}{r}\right)(-dt^2 + dr^{*2}) + r^2 d\Omega^2.
The thing about this radial coordinate is that the light cones don't close up anywhere. The downside, however, is that the surface r=2M has now been pushed to r^*\to-\infty. So, we introduce advanced and retarded Eddington-Finkelstein coordinates by defining
v \equiv t + r^*,
u \equiv t - r^*.
Then, for example, the metric in (u,r,\theta,\phi) coordinates becomes
ds^2 = -\left(1-\frac{2M}{r}\right)du^2 - 2dudr + r^2d\Omega^2.
(I think this is correct.) Now to my question. Is there any way that I can conformally compactify this metric so that I can draw a Penrose diagram of it? I know that I can go to Kruskal-Szekeres coordinates and construct a Penrose diagram there, but surely it's possible to construct Penrose diagrams using just (u,r,\theta,\phi) or (v,r,\theta,\phi) coordinates. My problem is that I can't readily see what coordinate transformations I should take so as to get to a finite coordinate range from the infinite ranges of u,v,r. Can anyone shed some light on this for me?
ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2.
The immediate difficulty with this is that if one looks at radial null curves one finds that the light cones appear to close up as r\to 2M. The standard way of getting around this is to define a tortoise radial coordinate according to
r^* \equiv r + 2M\log|r-2M|.
Then the metric in (t,r^*,\theta,\phi) coordinates becomes simply
ds^2 = \left(1-\frac{2M}{r}\right)(-dt^2 + dr^{*2}) + r^2 d\Omega^2.
The thing about this radial coordinate is that the light cones don't close up anywhere. The downside, however, is that the surface r=2M has now been pushed to r^*\to-\infty. So, we introduce advanced and retarded Eddington-Finkelstein coordinates by defining
v \equiv t + r^*,
u \equiv t - r^*.
Then, for example, the metric in (u,r,\theta,\phi) coordinates becomes
ds^2 = -\left(1-\frac{2M}{r}\right)du^2 - 2dudr + r^2d\Omega^2.
(I think this is correct.) Now to my question. Is there any way that I can conformally compactify this metric so that I can draw a Penrose diagram of it? I know that I can go to Kruskal-Szekeres coordinates and construct a Penrose diagram there, but surely it's possible to construct Penrose diagrams using just (u,r,\theta,\phi) or (v,r,\theta,\phi) coordinates. My problem is that I can't readily see what coordinate transformations I should take so as to get to a finite coordinate range from the infinite ranges of u,v,r. Can anyone shed some light on this for me?